4MA0 Higher January 2017 Paper 4 Q18
18 The function f is such that \(\quad\mathrm{f}(x) = \dfrac{2x}{3x + 5}\)
(a) Find \(\;\mathrm{f}(-2)\) (1)
The function g is such that \(\quad\mathrm{g}(x) = \dfrac{3}{x + 4}\)
(b) Find \(\;\mathrm{g}^{-1}(6)\) (2)
(c) Find \(\;\mathrm{fg}(-5)\) (2)
(d) Solve the equation \(\;\mathrm{f}(x) = \mathrm{g}(x)\)
Show clear algebraic working. (4)
Show clear algebraic working. (4)
| Scheme | Marks |
|---|---|
| 4 | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(6 = \dfrac{3}{x + 4}\) or (\(x =\)) \(\dfrac{3}{6} - 4\) or \(\dfrac{3 - 4 \times 6}{6}\) oe | M1 |
| \(-3\frac{1}{2}\) | A1 |
| (2) |
Notes
M1: or \(\left(\mathrm{g}^{-1}(x) =\right) \dfrac{3}{x} - 4\) or \(\dfrac{3 - 4x}{x}\)
A1: oe
| Scheme | Marks |
|---|---|
| f(−3) or \(\dfrac{2 \times -3}{3 \times -3 + 5}\) or \(\dfrac{2 \times \dfrac{3}{-5 + 4}}{3 \times \dfrac{3}{-5 + 4} + 5}\) | M1 |
| \(1\dfrac{1}{2}\) | A1 |
| (2) |
Notes
M1: Or for (g(−5) =) \(\dfrac{3}{-5 + 4}\) or −3
A1: or \(\dfrac{6}{4}\) or \(\dfrac{3}{2}\) or 1.5
| Scheme | Marks |
|---|---|
\(2x(x + 4) = 3(3x + 5)\) or \(2x^2 + 8x = 9x + 15\) oe or \(\dfrac{2x(x + 4)}{(3x + 5)(x + 4)} = \dfrac{3(3x + 5)}{(3x + 5)(x + 4)}\) or \(\dfrac{2x(x + 4)}{(3x + 5)(x + 4)} - \dfrac{3(3x + 5)}{(3x + 5)(x + 4)}\) (= 0) | M1 |
| \(2x^2 - x - 15\) (= 0) or \(\dfrac{2x^2 - x - 15}{(3x + 5)(x + 4)}\) (= 0) | A1 |
| \((2x + 5)(x - 3)\) (= 0) or \(\dfrac{(2x + 5)(x - 3)}{(3x + 5)(x + 4)}\) (= 0) | M1 |
| \(-2\frac{1}{2}\), 3 | A1 |
| (4) | |
| (9 marks) |
Notes
M1: or correct substitution into quadratic formula
or correctly completing the square
A1: dep on previous M1